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Q&A

How can school children intuit why over 100, D is larger? But under 100, D% is larger?

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I can prove the Rule of 100 algebraically, below. But my school kids are hankering after intuition, and a plainer explanation.

Follow the Rule of 100

Should discounts be percentages or absolutes?

Consider a $150 blender. Should you offer 20% off? Or an equivalent $30 off?

Answer:

  • **Over $100?** Give absolutes (e.g., $30)
  • Under $100? Give percents (e.g., 20%)

In both cases, you show the higher numeral. For a $50 blender, 20% off is the same as \10offyet2030 off) is a higher numeral (González, Esteva, Roggeveen, & Grewal, 2016).

References

González, E. M., Esteva, E., Roggeveen, A. L., & Grewal, D. (2016). Amount off versus percentage off—when does it matter?. Journal of Business Research, 69(3), 1022-1027.

Let d = discount, p = price. Then d,p>0 because there is no free lunch and the quotation is expatiating on discounts. Then

$ off vs. % off pd vs p(d/100)pd vs dp/1001 vs p/100p vs 100.

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x-post https://www.reddit.com/r/Precalculus/comments/1257bv0/how_can_school_children_intuit_why_if_d_... (1 comment)
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2 answers

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−1

If cost is c, sold for s, discount, d=cs

Discount percentage = csc×100=(cs)×100c

Absolute discount = cs

As you can see ...

  1. When c<100, 100c>1, and so, (cs)×100c>(cs). The percentage discount > The absolute discount.

  2. When c>100, 100c<1, and so (cs)×100c<(cs). The absolute discount > The percentage discount.

  3. When c=100, The absolute discount = The percentage discount. 100c=100100=1


EDIT 1 START

Intuitively, we begin where it starts (tautologies are always so out there), which is Discount (as a) percentage (of cost price) = csc×100, where c = cost price and s is the selling price.

Notice that csc×100=(cs)×100c. What we've achieved by doing this is that we can compare how the discount percentage varies with how c relates to 100.

When c=100, we get cs which is the absolute discount amount, which is also equal to the discount percentage.

When c<100,cs gets multiplied by an improper fraction 100c>1, which means the absolute discount amount cs will be less than the percentage discount csc×100 which we saw is the same as (cs)×100c.

When c>100, cs gets multiplied by an proper fraction 100c<1, which means the absolute discount amount cs will be greater than the percentage discount css×100, which we say is the same as (cs)×100c.

EDIT 1 END

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(I’m going to do this post in pounds so that I don’t have to escape dollar signs everywhere. But the currency doesn’t matter.)

“Per cent” is, literally, “for each hundred”. Imagine making a literal pile of money from the cost of the item, and splitting it up into stacks of £100. So if the item costs £500, you have five stacks.

Now, a discount that’s “ten pounds” means just that: take £10 out of the pile (any stack, it doesn’t matter) and put it back in your wallet.

But “ten per cent” means £10 out of every single one of those stacks. For five stacks, that’s £50 I get to keep! I’d rather have that discount.

What if the price is only £100, though? In that case, there’s only one stack, and “£10 from one stack” and “£10 from every stack” mean the same thing.

Below £100, the intuition is a little bit trickier, because you have to imagine the 10% case as taking “part of” £10 from a stack that’s “part of” £100. But it’s not an insurmountable hurdle to envision that, and grasp that it won’t be as good as taking out a full £10.

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